Topological characterizations of ordered groups with quasi-divisor theory

Jiří Močkoř

Czechoslovak Mathematical Journal (2002)

  • Volume: 52, Issue: 3, page 595-607
  • ISSN: 0011-4642

Abstract

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For an order embedding G → h → Γ of a partly ordered group G into an l -group Γ a topology 𝒯 W ^ is introduced on Γ which is defined by a family of valuations W on G . Some density properties of sets h ( G ) , h ( X t ) and ( h ( X t ) ∖ { h ( g 1 ) , ⋯ , h ( g n ) } ) ( X t being t -ideals in G ) in the topological space ( Γ , 𝒯 W ^ ) are then investigated, each of them being equivalent to the statement that h is a strong theory of quasi-divisors.

How to cite

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Močkoř, Jiří. "Topological characterizations of ordered groups with quasi-divisor theory." Czechoslovak Mathematical Journal 52.3 (2002): 595-607. <http://eudml.org/doc/30728>.

@article{Močkoř2002,
abstract = {For an order embedding $G\overset\{h\}\{\rightarrow \}\{\rightarrow \}\Gamma $ of a partly ordered group $G$ into an $l$-group $\Gamma $ a topology $\mathcal \{T\}_\{\widehat\{W\}\}$ is introduced on $\Gamma $ which is defined by a family of valuations $W$ on $G$. Some density properties of sets $h(G)$, $h(X_t)$ and $(h(X_t)\setminus \lbrace h(g_1),\dots ,h(g_n)\rbrace )$ ($X_t$ being $t$-ideals in $G$) in the topological space $(\Gamma ,\mathcal \{T\}_\{\widehat\{W\}\})$ are then investigated, each of them being equivalent to the statement that $h$ is a strong theory of quasi-divisors.},
author = {Močkoř, Jiří},
journal = {Czechoslovak Mathematical Journal},
keywords = {quasi-divisor theory; ordered group; valuations; $t$-ideal; quasi-divisor theory; ordered group; valuations; -ideal},
language = {eng},
number = {3},
pages = {595-607},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Topological characterizations of ordered groups with quasi-divisor theory},
url = {http://eudml.org/doc/30728},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Močkoř, Jiří
TI - Topological characterizations of ordered groups with quasi-divisor theory
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 595
EP - 607
AB - For an order embedding $G\overset{h}{\rightarrow }{\rightarrow }\Gamma $ of a partly ordered group $G$ into an $l$-group $\Gamma $ a topology $\mathcal {T}_{\widehat{W}}$ is introduced on $\Gamma $ which is defined by a family of valuations $W$ on $G$. Some density properties of sets $h(G)$, $h(X_t)$ and $(h(X_t)\setminus \lbrace h(g_1),\dots ,h(g_n)\rbrace )$ ($X_t$ being $t$-ideals in $G$) in the topological space $(\Gamma ,\mathcal {T}_{\widehat{W}})$ are then investigated, each of them being equivalent to the statement that $h$ is a strong theory of quasi-divisors.
LA - eng
KW - quasi-divisor theory; ordered group; valuations; $t$-ideal; quasi-divisor theory; ordered group; valuations; -ideal
UR - http://eudml.org/doc/30728
ER -

References

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  1. Ideale in kommutativen Halbgruppen, Rec. Math. Soc. Math. Moscow 36 (1929), 401–407. (German) (1929) 
  2. Lattice-ordered Groups, D. Reidl Publ. Co., Dordrecht, Tokyo, 1988. (1988) MR0937703
  3. Divisors of finite character, Ann. Mat. Pura Appl. 33 (1983), 327–361. (1983) Zbl0533.20034MR0725032
  4. Localizations dans les systémes d’idéaux, C. R. Acad. Sci. Paris 272 (1971), 465–468. (1971) MR0277511
  5. Number Theory, Academic Press, New York, 1966. (1966) MR0195803
  6. Lattice Ordered Groups, Tulane University, 1970. (1970) Zbl0258.06011
  7. Krull semigroups and divisor class group, Canad. J. Math. 33 (1981), 1459–1468. (1981) MR0645239
  8. 10.1016/0022-4049(94)00088-Z, J.  Pure Appl. Algebra 102 (1995), 289–311. (1995) MR1354993DOI10.1016/0022-4049(94)00088-Z
  9. Multiplicative Ideal Theory, M.  Dekker, Inc., New York, 1972. (1972) Zbl0248.13001MR0427289
  10. Rings of Krull type, J. Reine Angew. Math. 229 (1968), 1–27. (1968) Zbl0173.03504MR0220726
  11. Some results on v -multiplication rings, Canad. J. Math. 19 (1967), 710-722. (1967) Zbl0148.26701MR0215830
  12. Les systémes d’idéaux, Dunod, Paris, 1960. (1960) Zbl0101.27502MR0114810
  13. Groups of Divisibility, D. Reidl Publ.  Co., Dordrecht, 1983. (1983) MR0720862
  14. Approximation Theorems in Commutative Algebra, Kluwer Academic publ., Dordrecht, 1992. (1992) MR1207134
  15. Groups with quasi-divisor theory, Comm. Math. Univ. St. Pauli, Tokyo 42 (1993), 23–36. (1993) MR1223185
  16. Divisor class groups of ordered subgroups, Acta Math. Inform. Univ. Ostraviensis 1 (1993), 37–46. (1993) MR1250925
  17. Quasi-divisors theory of partly ordered groups, Grazer Math. Ber. 318 (1992), 81–98. (1992) MR1227404
  18. 10.1016/S0022-4049(96)00059-X, J. Pure Appl. Algebra 120 (1997), 51–65. (1997) MR1466097DOI10.1016/S0022-4049(96)00059-X
  19. Some remarks on Lorezen r -group of partly ordered group, Czechoslovak Math.  J. 46(121) (1996), 537–552. (1996) MR1408304
  20. Divisor class group and the theory of quasi-divisors, To appear. MR1765996
  21. Semi-valuations and groups of divisibility, Canad. J. Math. 21 (1969), 576-591. (1969) Zbl0177.06501MR0242819
  22. Divisorentheorie einer Halbgruppe, Math.  Z. 114 (1970), 113–120. (1970) Zbl0177.03202MR0262401
  23. 10.4064/aa-31-3-247-257, Acta Arith. 31 (1976), 247–257. (1976) Zbl0303.13014MR0444817DOI10.4064/aa-31-3-247-257

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